Far-Field Interference of a Neutron White Beam and the Applications To

PHYSICAL REVIEW A 95, 043637 (2017)

Far-ﬁeld interference of a neutron white beam and the applications to noninvasive phase-contrast imaging

D. A. Pushin,1,2,* D. Sarenac,1,2 D. S. Hussey,3 H. Miao,4 M. Arif,3 D. G. Cory,2,5,6,7 M. G. Huber,3 D. L. Jacobson,3 J. M. LaManna,3 J. D. Parker,8 T. Shinohara,9 W. Ueno,9 and H. Wen4 1Department of Physics, University of Waterloo, Waterloo, Ontario, Canada N2L3G1 2Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L3G1 3National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 4Biophysics and Biochemistry Center, National Heart, Lung and Blood Institute, National Institutes of Health, Bethesda, Maryland 20892, USA 5Department of Chemistry, University of Waterloo, Waterloo, Ontario, Canada N2L3G1 6Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L2Y5 7Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8 8Research and Development Division, Neutron Science and Technology Center, Comprehensive Research Organization for Science and Society (CROSS), 162-1 Shirakata, Tokai, Ibaraki 319-1106, Japan 9Materials and Life Science Division, J-PARC Center Japan Atomic Energy Agency (JAEA), 2-4 Shirakata, Tokai, Ibaraki 319-1195, Japan (Received 19 September 2016; revised manuscript received 13 December 2016; published 26 April 2017) The phenomenon of interference plays a crucial role in the field of precision measurement science. Wave- particle duality has expanded the well-known interference effects of electromagnetic waves to massive particles. The majority of the wave-particle interference experiments require a near monochromatic beam which limits its applications due to the resulting low intensity. Here we demonstrate white beam interference in the far-field regime using a two-phase-grating neutron interferometer and its application to phase-contrast imaging. The functionality of this interferometer is based on the universal moiré effect that allows us to improve upon the standard Lau setup. Interference fringes were observed with monochromatic and polychromatic neutron beams for both continuous and pulsed beams. Far-field neutron interferometry allows for the full utilization of intense neutron sources for precision measurements of gradient fields. It also overcomes the alignment, stability, and fabrication challenges associated with the more familiar perfect-crystal neutron interferometer, as well as avoids the loss of intensity due to the absorption analyzer grating requirement in Talbot-Lau interferometer.

DOI: 10.1103/PhysRevA.95.043637

I. INTRODUCTION later, a three-transmission-phase-grating Mach-Zehnder NI was demonstrated for cold neutrons (0.2 nm <λ<50 nm) The discovery of the neutron [1] led to the construction of a [13–16]. The need for cold or very cold neutrons with a variety of phase-sensitive neutron interferometers. Thermal high degree of collimation limits the use of such interfer- and cold neutrons are a particularly convenient probe of ometers in material science and condensed-matter research. matter and quantum mechanics postulates given their relatively An alternative approach was the Talbot-Lau interferometer large mass, nanometer-sized wavelengths, and zero electric (TLI) proposed by Clauser and Li for cold potassium ions charge. One of the first neutron interferometers [2] was based and x-ray interferometry [17] and implemented by Pfeiffer on wave-front division using a Fresnel biprism setup. The et al. for neutrons [18]. The TLI is based on the near-field perfect-crystal neutron interferometer (NI), based on ampli- Talbot effect [19] and uses a combination of absorption and tude division, has achieved the most success due to its size phase gratings. In this setup the sample, introduced in front and modest path separation of a few centimeters. Numerous of the phase grating (middle grating), modifies the phase perfect-crystal NI experiments have been performed exploring and amplitude of the Talbot self-image. While the previously the nature of the neutron and its interactions [3–9]. However, mentioned Mach-Zehnder-type grating interferometers are perfect-crystal neutron interferometry requires extreme forms sensitive to phase shifts induced by a sample located in one of environmental isolation [10,11], which significantly limits arm of the interferometer, this near-field TLI is sensitive to its expansion and development. phase gradients caused by a sample. Even though chromatic Advances in micro- and nanofabrication of periodic struc- sensitivity of the TLI is reduced, thus leading to a gain tures with features ranging between 1 and 100 μm makes in neutron intensity, in this setup the absorption gratings it possible to employ absorption and phase gratings as are challenging to manufacture and limits the incident flux practical optical components for neutron beams. The first reaching the detector; while the neutron wavelength spread demonstration of a Mach-Zehnder-based grating NI in 1985 will cause contrast loss as the distance to interference fringes [12] used 21-μm periodic reflection gratings as beam splitters (fractional Talbot distance) is inversely proportional to the for monochromatic (λ = 0.315 nm) neutrons. A few years neutron wavelength. It has been previously demonstrated for atoms and electromagnetic waves that the two-grating TLI setup produces *[email protected] fringes in the far field [20–23]. This “Lau effect” [24] requires

the first grating to be an intensity mask which serves as an L array of mutually incoherent sources, and the second grating 1 Incident Slit D is either an intensity or a phase mask which through Fresnel Beam L diffraction results in the interference pattern on the screen. 2 When the distance between the two gratings is at multiples of half Talbot length, an image of the first grating is produced Sample at the plane at the second grating, and that image beats with G the second grating to produce the beat pattern in the far field x 1 G downstream. 2 y z Here we implement a far-field regime interferometer and report the first demonstration of a multibeam, broadband Camera NI using exclusively phase gratings. Both Lau and Talbot effects turned out to be special cases of a more universal FIG. 1. Schematic of the two-phase-grating interferometer setup. moiré effect [25]. The conceptual step forward of our neutron A neutron beam is passed through a narrow slit to define the neutron interferometer is to go to the other end of the moiré effect coherence length along the y direction, which is perpendicular to spectrum for the matter waves, where two pure phase masks the grating fringes. Two identical phase gratings (G1 and G2)are produce a beat pattern when both the source and the detector separated by distance D. They are placed at a distance L1 from the are in the far-field range. This interference would be observed slit and L2 from the imaging camera. The fringe pattern at the camera even if the two phase gratings are in contact, provided that is wavelength independent and the fringe visibility can be optimized their periods are appropriately different. The demonstration with the conditions discussed in the text. A sample to be imaged may includes the use of a continuous monochromatic beam (λ = be placed before the gratings (upstream position) or after the gratings 0.44 nm), a continuous bichromatic beam (1/3 intensity (downstream position). λ1 = 0.22 nm and 2/3 intensity λ2 = 0.44 nm), a continuous polychromatic beam (approximately given by a Maxwell- In order for the neutron to diffract from the first grating G1 Boltzmann distribution with Tc = 40 K or λc = 0.5nm), at the distance L1, the neutron’s coherence length (along the and a pulsed neutron beam (λ = 0.5nmtoλ = 3.5nm). y axis in Fig. 1) should be at least equal to the period of the The advantages of this setup include the use of widely grating: available thermal and cold neutron beams, relaxed grating λL1 fabrication and alignment requirements, and broad wavelength = P . (1) c s G1 acceptance. w The second grating G2 is placed at a distance D from the first grating and a distance L2 from the neutron camera. As II. TWO-PHASE-GRATING INTERFEROMETER neutron cameras have limited spatial resolution η, the fringe As the neutron can be described as a matter wave with a de period Pd at the camera should be bigger than the neutron Broglie wavelength λ = 2πh/¯ (mnvn), whereh ¯ is the reduced camera resolution [25], Planck’s constant, mn is the neutron mass, and vn is the neutron LP P = G2 >η, (2) velocity, the problem could be treated similar to the x-ray case. d D The full mathematical treatment of the general situation of a L = L + D + L polychromatic beam passing through a double-phase-grating where 1 2 is the distance between the slit and setup (see Fig. 1) is described by Miao et al. [25]. Here we give the camera. Similarly, the phase of the fringe pattern on the a brief description and the key points of the universal moiré detector is a periodic function of the slit position, with the effect in a far-field regime for neutrons. period (often called source period) given by [25] The experimental setup, shown in Fig. 1, consists of a LP P = G1 . (3) slit, two identical linear phase gratings of silicon combs with s D a period of P = P = 2.4 μm, and a neutron imaging G1 G2 Therefore, in order to observe a fringe pattern on the detector (neutron camera). Although π/2-phase gratings for detector, the slit width should be smaller than the source period; a specific neutron wavelength give optimal fringe visibility, i.e., Ps >sw. available to us were five different gratings with various depths. To verify that we are indeed in a far-field regime, we We had used gratings corresponding to 0.27π phase shift at consider the Fraunhofer distance when the coherence length is 0.44-nm wavelength for the mono- and bichromatic setups and used as the source dimension: 0.2π phase shift at 0.5 nm wavelength for the polychromatic 2 2 setup. c L1 dF = 2 = 2λ . (4) For a majority of this work, a fast neutron produced in a λ sw reactor core is first moderated using heavy water to thermal We consider the coherence length because it is always equal energies and then further cooled using a liquid hydrogen cold to or greater than the grating period in the setup. To satisfy source [26] before traversing a neutron guide, the end of which the far-field regime, L2 should be greater than the Fraunhofer is a slit. After exiting the slit and propagating in free space, distance: the neutron acquires a transverse coherence length of c = dF λL λL /sw, where sw is the width of the slit and L is the distance ≈ 1. (5) 2 between the slit and the point of interest. L2 sw

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Given the experimental parameters of the monochromatic beamline dF /L2 ≈ 0.04, polychromatic beamline dF /L2 ≈ 0.4 0.02 for λ = 0.5 nm, and the beamline at the pulsed source d /L ≈ 0.08 for λ = 0.35 nm. The other two conditions for F 2 0.3 far-ﬁeld regime are

L2 λ, L2 sw. (6) 0.2 Contrast In our cases they are the least strict conditions. = ≡ 0.1 If we consider equal period PG1 PG2 Pg, π/2-phase gratings, with 50% comb fraction, then the maximum contrast is optimized for the condition δ (λ) = δ (λ) = 0.5[25], where 0.0 1 2 -10 -5 0 5 10 15×10-3 λL1D λL2D G1 Rotation [deg] δ (λ) = ,δ(λ) = . (7) 1 2 2 2 LPg LPg 0.4 The closed-form expression of the contrast is given by Eq. 12 in [25] and is computed numerically. 0.3 To align the gratings, the setup is initially arranged with theoretically calculated optimal slit width and lengths L1, D, and L2. Then one of the gratings is rotated around the neutron 0.2 propagation axis (z axis) until the fringe pattern is observed at Contrast the camera. The contrast with the monochromatic setup as a 0.1 function of the first grating rotation around the z axis is shown in the top plot of Fig. 2. The slit height s (slit length along the x-axis direction in 0.0 h 0 200 400 600 800 Fig. 1) can be larger than the slit width sw in order to increase neutron intensity, provided that the gratings are well aligned to Slit Width [µm] be parallel to that direction. The angular range of appreciable FIG. 2. Optimization data. (Top) The contrast is found by placing contrast is inversely related to the slit height Pg/(2L1sh/L). The expected range of ±0.007◦ agrees with the range depicted the gratings vertically and rotating one of them along the roll on Fig. 2. axis in very fine increments. The plot shows the contrast at the The intensity of the fringe pattern recorded by the camera monochromatic beamline as a function of the first grating rotation around the neutron propagation axis. (Bottom) The dependence of can be fit by a cosine function the contrast at the bichromatic setup on the slit width sw.Thefitis given by applying Eq. (10) to the data and shows good agreement I = A + B cos(fx+ φ). (8) with the calculated source period Ps . where x is the pixel location on the camera. Thus, the mean A, the amplitude B, the frequency f , and the differential phase φ III. METHODS contrast fringe visibility can be extracted from the fit. The ,or , The experiment was performed in four different configu- is given by rations. The bichromatic and monochromatic beam configu- max{I}−min{I} B rations were performed at the NG7 NIOFa beamline [27]at C = = . (9) the National Institute of Standards and Technology Center for max{I}+min{I} A Neutron Research (NCNR) with L1 = 1.73 m and L = 3.52 m = = Due to the generally low neutron flux with monochromatic for bichromatic beam and L1 1.20 m and L 2.99 m for beamlines, the slit widths are optimized for intensity vs monochromatic beam. The neutron camera used in this setup has an active area of 25 mm diameter, with scintillator NE426 contrast. The contrast as a function of the slit width for the 6 bichromatic setup is shown in the bottom plot of Fig. 2. [ZnS(Ag) type with Li as the neutron converter material] and ∼ Variation of the contrast vs slit width could be described by a spatial resolution of 100 μm, and virtually no dark current the sinc function noise [28]. Images were collected in 300-s-long exposures. The neutron quantum efficiency of the camera is 18% for λ = πs 0.22 nm and about 50% for λ = 0.44 nm. The neutron beam is C = C0sinc , (10) Ps extracted from a cold neutron guide by a pyrolytic graphite (PG) monochromator with λ = 0.44 nm and λ = 0.22 nm where C0 is the maximum achievable contrast with a given components with an approximate ratio of 3.2:1 in intensity. To setup. Thus, given a slit width of one-third of the source change from bichromatic to monochromatic configuration, i.e., period, in our case 281 μm, would give an upper bound of filter out the λ = 0.22 nm component, a liquid-nitrogen-cooled 83% contrast. The fit in the bottom plot of Fig. 2 gives a source Be filter with nearly 100% filter efficiency [27] was installed period of Ps = 843 ± 43 μm, which is in good agreement with downstream of the interferometer entrance slit. The slit width Ps = 845 μm obtained with Eq. (3), where D = 10 mm. was set to 200 μm and slit height to 2.5 cm.

043637-3 D. A. PUSHIN et al. PHYSICAL REVIEW A 95, 043637 (2017)

The polychromatic beam configuration was performed at (a) Bichromatic (b) Monochromatic the NG6 Cold Neutron Imaging (CNI) facility [29]atthe NCNR. The CNI is located on the NG6 end station and has neutron spectrum approximately given by a Maxwell- Boltzmann distribution with Tc = 40 K or λc = 0.5nm.The slit-to-detector length is L = 8.8 m and slit to G1 distance is L1 = 4.65 m. The slit width was set to 500 μm and slit height to 1.5 cm. The imaging detector was an Andor sCMOS NEO camera viewing a 150-μm-thick LiF:ZnS scintillator with a Nikon 85-mm lens with a PK12 extension tube for a reproduction ratio of about 3.7, yielding a spatial resolution of η = 150 μm [30]. To reduce noise in the sCMOS system, the median of three images was used for analysis. 1500 The fourth configuration used a pulsed neutron beam 12000 produced at the Energy-Resolved Neutron Imaging System 1000 8000 (RADEN) [31], located at beam line BL22 of the Japan Proton Accelerator Research Complex (J-PARC) Materials 500 4000 Avg Intensity Avg Avg Intensity Avg 0 0 and Life Science Experimental Facility (MLF). The wave- 0 2 4 6 8 10 0 1 2 3 4 5 6 7 length range that was used was from 0.05 to 0.35 nm. The Distance [mm] Distance [mm] slit-to-detector length is L = 8.6 m and slit-to-G1 distance (c) Polychromatic (d) Pulsed Source is L1 = 4.24 m. The slit width was set to 200 μm and slit height to 4 cm. The neutron imaging system employed a micropixel chamber (μPIC), a type of micropattern gaseous detector with a two-dimensional strip readout, coupled with an all-digital, high-speed field-programmable gate array (FPGA)- based data-acquisition system [32]. This event-type detector records the time of arrival of each neutron event relative to the pulse start time for precise measurement of neutron energy, and it has a spatial resolution of 280 μm(FWHM). The readout of the μPIC detector introduces a fixed-pattern noise structure which is completely removed by normal- izing to empty beam measurements. Thus, the visibility measurements are from open-beam normalized images of the moiré pattern. The average number of detected neutron events was about 80 per 160-μm pixel with a 4-h integration 100 1.10 80 1.05 time. 60 40 1.00 20 0.95

IV. RESULTS AND DISCUSSION Intensity Avg 0 Intensity Avg 0 5 10 15 20 25 0 5 10 15 20 By observing the interference fringes with and without a Distance [mm] Distance [mm] sample that has been placed either upstream or downstream of the gratings, we can extract the conventional beam attenuation, FIG. 3. Typical far-field images obtained with (a) 300-s exposure dark-field contrast, and differential phase contrast. By varying time with the bichromatic neutron beam, (b) 10000-s exposure time the grating spacing D, this interferometer may also be with the monochromatic neutron beam, and (c) median filter applied employed to measure small-angle scattering of microstructures on three images with 2-s exposure time with the polychromatic in the range of nm to several μm[33]. neutron beam; (d) normalized image from the J-PARC pulsed source. Figures 3(a)–3(d) show examples of typical images of the The integrated intensities of the regions specified by the yellow rectangle show the observed fringe pattern at the camera. The dark interference pattern obtained in optimized configurations for region in the middle of the bichromatic image is due to the collimator different beamlines: (a) bichromatic beam with λ = 0.22 nm 1 in that particular setup and not due to the gratings. and λ2 = 0.44 nm (b) same beamline as (a) but with a Be filter to completely eliminate the λ1 = 0.22-nm-component (c) polychromatic neutron beam with peak wavelength λc = Such integral curves were used to fit with Eq. [8] to extract 0.5 nm and (d) a pulsed source for λ = 0.25 nm. In Fig. 3(a) phase and frequency and compute the contrast via Eq. [9]. the middle dark region corresponds to the collimator which The top plot in Fig. 4 shows contrast (fringe visibility) was placed at the front in the setup and not the grating pattern. change versus grating separation, D. The data obtained at the As the Be filter adds divergence to the beam, it can be seen National Institute of Standards and Technology (NIST) for the that the dark middle region gets washed out in Fig. 3(b).The monochromatic, bichromatic and polychromatic beamlines are box on each image represents a region of integration along the plotted on the same figure for comparison. The theoretically vertical axis and the integral curve is shown under each image. calculated contrast curves for the three conditions are also

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0.5 Monochromatic Bichromatic 0.4 Polychromatic

0.3 Contrast 0.2

0.1 8 10 12 14 16 Separation Distance D [mm] 0.8 Monochromatic ]

-1 0.7 Bichromatic Polychromatic 0.6 0.5 0.4 0.3 Frequency [mm Frequency 0.2 0.1 8 10 12 14 16 Separation Distance D [mm] FIG. 5. Linear phase stepping is achieved with parallel translation FIG. 4. (Top) The contrast as a function of the separation of the first grating G1 by increments smaller than the period distance between the gratings for the monochromatic, bichromatic, of the gratings. (Top) Data from the monochromatic beamline. and polychromatic neutron beams. The fits are given by Eq. 12 (Bottom) Data from the polychromatic beamline. In both cases linear in [25], and the contrast for the monochromatic, bichromatic, and dependence between the phase and grating translations is observed, = polychromatic beamlines is optimized at D 12, 12, and 10 mm, while the contrast is preserved. Data were collected for the translation respectively, agreeing well with theoretical predictions [25]. (Bottom) range of 0 to 5 μm, in increments of 0.2 μm (0.5 μm) for the The frequency of the oscillation fringes varies linearly as a function monochromatic (polychromatic) plot. of the distance between the gratings. The linear fit is according to (D − D0)/(L × Pg), which for the monochromatic setup gives D0 =−0.75 mm and L = 3.04 m, the bichromatic setup gives D0 =−0.8mmandL = 3.51 m, and polychromatic setup gives in Fig. 1 or along the grating vector). The translation step D0 =−3.8mmandL = 8.36 m. The exposure time was 2 s. size needs to be smaller than the grating period. The top plot in Fig. 5 shows the two-dimensional plot of phase stepping for the monochromatic beamline setup; and the bottom plot plotted, which were based on estimates of 0.27π phase- in Fig. 5 shows the phase stepping for the polychromatic shift gratings at 0.44 nm wavelength for the mono- and beamline setup. In both cases linear dependence between the bichromatic setups and 0.2π phase shift at 0.5 nm wavelength phase and grating translation is observed, while the contrast is for the polychromatic setup. The maximum contrast for the preserved. monochromatic, bichromatic, and polychromatic beamlines Similar aligning procedures and measurements were per- are achieved at D = 12, 12, and 10 mm, respectively, agreeing formed at J-PARC spallation pulsed source. Figure 6 shows well with theoretical predictions [25]. Theoretical estimates the contrast as a function of the wavelength for various grating indicate that there is room for improvement of contrast by separations. Due to the nature of the pulse source, we were improving grating profile and detector resolution. The bottom able to extract contrast as a function of wavelength. Note plot in Fig. 4 shows the linear dependence of the fringe that at the time of the experiments J-PARC was running at frequency at the camera on the grating separation. As the 200 KW as opposed to 1 MW due to technical problems. distance between the gratings is increased, the period of the This lowered the neutron flux to 1/5 of the standard flux fringes at the camera is decreased. The linear fit is according to and the low intensity proved to be a significant challenge (D − D0)/(L × Pg), which for the monochromatic setup gives in terms of optimizing the setup for each independent D0 =−0.75 mm and L = 3.04 m, the bichromatic setup gives wavelength. D0 =−0.8 mm and L = 3.51 m, and polychromatic setup The beam attenuation, decoherence, and phase-gradient gives D0 =−3.8 mm and L = 8.36 m. images shown in Figs. 7(b)–7(d) are of an aluminum sample To implement phase stepping of the fringe pattern at the shown in Fig. 7(a). The approximate imaged area is depicted camera, one grating needs to be translated in plane in the by the rectangular box. The images in Fig. 7(b)–7(d) were perpendicular direction to the grating lines (along the y axis obtained by the Fourier transform method described in [34].

043637-5 D. A. PUSHIN et al. PHYSICAL REVIEW A 95, 043637 (2017)

transmission (no attenuation). The shape and features of the sample are well deﬁned in the image. Figure 7(c) shows the decoherence of the fringe contrast due to the sample, − ln(Csample/Cempty), where the white color represents loss of contrast and the black color represent no contrast reduction. As expected, the areas which caused the largest attenuation also caused the largest loss of contrast, likely due to small angle scattering from alloying. Figure 7(d) shows the phase shift in the moiré pattern at the detector due to the sample. The white and black patterns represent highest phase gradient that the neutrons acquire when passing through the sample.

V. CONCLUSION For the first time we have demonstrated a functioning two-phase-grating-based, moiré-effect neutron interferometer. The design has a broad wavelength acceptance and requires nonrigorous alignment. The interferometer operates in the far- FIG. 6. Grayscale representation of the contrast (fringe visibility) field regime and can potentially circumvent many limitations dependence on neutron wavelength and gratings separation distance, D. Contrast data obtained with a pulsed neutron beam from a of the single crystal and grating-based Mach-Zehnder-type spallation source at J-PARC. Data were collected for the wavelength interferometers and the near-field Talbot-Lau-type interferom- range of 0.05 to 0.35 nm, in increments of 0.05 nm, and for separation eters that are in operation today. Mach-Zehnder-type interfer- distance values of 3, 6, 12, 18, and 24 mm. ometers may provide the most precise and sensitive mode of measurements, but a successful implementation requires highly collimated and low-energy neutron beams. On the other At the described polychromatic beamline at NIST three images hand, a near-field Talbot-Lau interferometer requires absorp- with 20-s exposure time were taken at each step in the tion analyzer gratings which curtail flux and interference fringe phase-stepping method [35]. A median filter was then applied contrast. These constraints can be significant in a variety of to every set of three images. The phase step size was 0.24 μm, applications. ranging from 0 to 2.4 μmoftheG2 transverse translation. The The performance of our demonstration interferometer = G1-G2 grating separation was D 11.5 mm. was limited primarily by grating imperfections and detector Figure 7(b) shows the conventional attenuation-contrast resolution. However, the design is simple and robust. We radiography of the sample, where white color represents full expect that the next generation of interferometers based on the far-field design will open new opportunities in high-precision phase-based measurements in materials science, condensed- (a) 1.0 (b) matter physics, and bioscience research. In particular, because 0.8 of the moiré fringe exploitation in this type of interferometers, the uses may be highly suitable for the studies of biological 0.6 membranes, polymer thin films, and materials structure. 0.4 Also, the modest cost and the simplicity of assembly and (c) operation will allow this type of interferometer to have wide 4 (d) 3 acceptance in small to modest research reactor facilities 2 3 worldwide. 1

2 0 -1 1 -2 ACKNOWLEDGMENTS 0 -3 This work was supported by the U.S. Department of Com- merce, the NIST Radiation Physics Division, the director’s FIG. 7. Phase-contrast imaging with a polychromatic beam. office of NIST, the NIST Center for Neutron Research, and the (a) The aluminum sample in front of the G1 grating. The yellow box roughly outlines what is being imaged. The sample has a step profile in National Institute of Standards and Technology (NIST) Quan- the middle region and threaded screw holes in the corners. (b) Neutron tum Information Program. The research results communicated attenuation image due to absorption and scattering. The grayscale bar here would not be possible without the significant contribu- represents transmission through the sample. (c) Spatial variation of tions of the Canada First Research Excellence Fund (CFREF), the contrast attenuation due to the sample, − ln(Csample/Cempty). In this the Canadian Excellence Research Chairs (CERC) program case white represents a loss in contrast, and black represents no loss (215284), the Natural Sciences and Engineering Research in contrast. (d) Phase shift in the moiré pattern at the detector due to Council of Canada (NSERC) Discovery program, and the the sample. The grayscale bar represents radians. Here, the white and Collaborative Research and Training Experience (CREATE) black patterns represent the highest phase gradient that the neutrons program (414061). D.A.P. is grateful for discussions with acquire when passing through the sample. Michael Slutsky.

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[1] J. Chadwick, Nature (London) 129, 312 (1932). [19] H. Talbot, Philos. Mag. Ser. 3 9, 401 (1836). [2] H. Maier-Leibnitz and T. Springer, Z. Physik 167, 386 (1962). [20] A. D. Cronin and B. McMorran, Phys.Rev.A74, 061602 (2006). [3] H. Rauch and S. A. Werner, Neutron Interferometry: Lessons in [21] C. David, B. Nöhammer, H. Solak, and E. Ziegler, Appl. Phys. Experimental Quantum Mechanics, Wave-Particle Duality, and Lett. 81, 3287 (2002). Entanglement, 2nd ed. (Oxford University Press, Oxford, UK, [22] A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, Rev. Mod. 2015), Vol. 12. Phys. 81, 1051 (2009). [4] C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, [23] M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmied- and D. A. Pushin, Nature (London) 525, 504 (2015). mayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, Phys. [5] K. Li, M. Arif, D. Cory, R. Haun, B. Heacock, M. Huber, J. Rev. A 51, R14 (1995). Nsofini, D. Pushin, P. Saggu, D. Sarenac et al., Phys. Rev. D 93, [24] E. Lau, Ann. Phys. 437, 417 (1948). 062001 (2016). [25] H. Miao, A. Panna, A. A. Gomella, E. E. Bennett, S. Znati, L. [6] D. A. Pushin, M. Arif, and D. G. Cory, Phys. Rev. A 79, 053635 Chen, and H. Wen, Nat. Phys. 12, 830 (2016). (2009). [26] R. E. Williams and J. M. Rowe, Physica B 311, 117 (2002). [7] D. Sarenac, M. G. Huber, B. Heacock, M. Arif, C. W. Clark, [27] C. Shahi, M. Arif, D. Cory, T. Mineeva, J. Nsofini, D. Sarenac, D. G. Cory, C. B. Shahi, and D. A. Pushin, Opt. Express 24, C. Williams, M. Huber, and D. Pushin, Nucl. Instrum. Methods 22528 (2016). Phys. Res., Sect. A 813, 111 (2016). [8] J. Klepp, S. Sponar, and Y. Hasegawa, Prog. Theor. Exp. Phys. [28] M. Dietze, J. Felber, K. Raum, and C. Rausch, Nucl. Instrum. 2014, 82A01 (2014). Methods Phys. Res., Sect. A 377, 320 (1996). [9] T. Denkmayr, H. Geppert, S. Sponar, H. Lemmel, A. Matzkin, [29] D. Hussey, C. Brocker, J. Cook, D. Jacobson, T. Gentile, W. J. Tollaksen, and Y. Hasegawa, Nat. Commun. 5, 4492 (2014). Chen, E. Baltic, D. Baxter, J. Doskow, and M. Arif, Phys. Proc. [10] M. Arif, D. E. Brown, G. L. Greene, R. Clothier, and K. Littrell, 69, 48 (2015). Proc. SPIE 2264, 20 (1994). [30] Certain trade names and company products are mentioned in the [11] P. Saggu, T. Mineeva, M. Arif, D. Cory, R. Haun, B. Heacock, text or identified in an illustration in order to adequately specify M. Huber, K. Li, J. Nsofini, D. Sarenac et al., Rev. Sci. Instrum. the experimental procedure and equipment used. In no case 87, 123507 (2016). does such identification imply recommendation or endorsement [12] A. I. Ioffe, V. S. Zabiyakin, and G. M. Drabkin, Phys. Lett. 111, by the National Institute of Standards and Technology; nor does 373 (1985). it imply that the products are necessarily the best available for [13] M. Gruber, K. Eder, A. Zeilinger, R. Gáhler, and W. Mampe, the purpose. Phys. Lett. A 140, 363 (1989). [31] T. Shinohara and T. Kai, Neutron News 26, 11 (2015). [14] G. van der Zouw, M. Weber, J. Felber, R. Gähler, P. Geltenbort, [32] J. Parker, K. Hattori, H. Fujioka, M. Harada, S. Iwaki, S. Kabuki, and A. Zeilinger, Nucl. Instrum. Methods Phys. Res., Sect. A Y. Kishimoto, H. Kubo, S. Kurosawa, K. Miuchi et al., Nucl. 440, 568 (2000). Instrum. Methods Phys. Res., Sect. A 697, 23 (2013). [15] U. Schellhorn, R. A. Rupp, S. Breer, and R. P. May, Phys. B [33] D. S. Hussey, H. Miao, G. Yuan, D. Pushin, D. Sarenac, Condens. Matter 234–236, 1068 (1997). M. G. Huber, D. L. Jacobson, J. M. LaManna, and H. Wen, [16] J. Klepp, C. Pruner, Y. Tomita, C. Plonka-Spehr, P. Geltenbort, arXiv:1606.03054. S. Ivanov, G. Manzin, K. H. Andersen, J. Kohlbrecher, M. A. [34] H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Car- Ellabban, and M. Fally, Phys.Rev.A84, 013621 (2011). roll, Medical Imaging, IEEE Trans. Med. Imaging 27, 997 [17] J. F. Clauser and S. Li, Phys.Rev.A49, R2213 (1994). (2008). [18] F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and [35] J. H. Bruning, D. R. Herriott, J. Gallagher, D. Rosenfeld, A. C. David, Phys. Rev. Lett. 96, 215505 (2006). White, and D. Brangaccio, Appl. Opt. 13, 2693 (1974).

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